Insuring Precision in Calculations
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When performing calculations with measured quantities, the
precision of the result is limited by the least precise measurement.
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The results of calculations involving measurements must be
rounded off to match the precision of the least precise measurement.
Rounding off
Limiting the number of digits in a measurement to
a specified number of significant figures.
Rules for Rounding Off
1. If the first digit to be dropped is 4 or less,
it and all following digits are simply dropped from the number.
2. If the first digit to be dropped is 5 or greater,
the last retained digit of the number is increased by 1.
Examples:
|
|
| Three Significant Figures |
|
|
| 8.4234 |
8.42 |
8.4 |
|
|
|
| 14.780 |
14.8 |
15 |
|
|
|
| 3,256 |
3,260 |
3,300 or 3.3 x 103 |
Very Basic Steps for Rounding Off:
1. Underline the digit in the place you are rounding
to.
2. Circle the digit to the right of the underlined digit.
3. If the circled digit is four or less, the underlined
digit stays the same.
If the circled digit is five or more, add one to
the underlined digit.
4. All the digits to the left of the underlined digit
stay the same.
5. Finally, all the digits to the right of the underlined
digit go away.
Why do measurements and calculations
need to be rounded off?
Practice Problems
Answer the following questions: |
|
| 1. |
Round off each of the following numbers
1. to three significant
figures
2. to two significant
figures: |
|
.
| a) 35.7823 m |
b) 0.002 627 L |
c) 3.8268 x 103
g |
d) 1.2836 kg |
|
|
|
Significant Figures in Multiplication
and Division
In multiplication or division, the final answer
is written so it has the same number of significant figures as the measurement
with the fewest significant figures (SFs).
Example 1: Multiplication:
|
24.65
|
x
|
0.67
|
=
|
16.5155
|
>
|
17
|
|
(4 SF’s)
|
|
(2 SF’s)
|
|
|
round
|
(2 SF’s)
|
0.67 has the fewest significant figures—two—so
the final answer has to have the same number of significan figures—two.
Example 2: Multiplication and Division
|
2.85
|
x
|
67.4
|
÷
|
4.39
|
=
|
43.756264
|
>
|
43.8
|
|
(3 SF’s)
|
|
(3 SF’s)
|
|
(3 SF’s)
|
|
calculator
|
round
|
(3 SF’s)
|
IMPORTANT: Do not round off until you have calculated
the final answer!
How many significant figures
should the final answer have?
Practice Problems
Answer the following questions: |
|
| 2. |
Work the problems and then write the answers
in the correct number of significant digits: |
|
.
| a) 56.8 x 0.37 |
b) 71.4 ÷ 11 |
c) 25.0 ÷ 5.00 |
| . |
|
|
|
d) 2.075 x 0.585 ÷ (8.42 x 0.0045) |
|
|
|
Addition and Subtraction
Significant figures translate
into DECIMAL PLACES in
Addition and Subtraction
In addition or subtraction, the final answer is written
so it has the same number of decimal places as the measurement with the
fewest
decimal places (DP’s).
Example 3: Addition and Subtraction:
Remember: Don’t count significant figures—count
digits
after the decimal.
| Add |
|
|
Subtract |
|
|
|
2.045 |
(3 DPs) |
|
14.51 |
(2 DPs) |
|
+ 34.1 |
(1 DP) |
|
- 2.5 |
(1 DP) |
|
36.145 |
|
|
12.01 |
|
| . |
|
round off |
|
|
round off |
|
36.1 |
(1 DP) |
|
12.0 |
(1 DP) |
What do addition and subtraction
use instead of decimal places?
Practice Problems
Answer the following questions: |
|
| 3. |
State the answers with the correct number
of decimal places: |
|
.
| a) 27.8 cm + 0.235 cm |
b) b) 153.247 g – 14.82 g |
|
|
|
|
